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Excitonic and ultraexcitonic effects in supramolecular architectures of polar and polarizable chromophores
Synthetic Metals 139 (2003) 779–781
Excitonic and ultraexcitonic effects in supramolecular architectures of polar and polarizable chromophores Anna Painelli∗ , Francesca Terenziani Dip. Chimica GIAF, Università di Parma, viale delle Scienze 17/a, 43100 Parma, Italy, and INSTM-UdR Parma
Abstract Supramolecular interactions in clusters of polar and polarizable molecules are discussed. With reference to two different one-dimensional arrays of molecules we investigate the dependence of the molecular polarity and of static non-linear optical responses on the intermolecular distance. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Organic semiconductors based on conjugated molecules; Semi-empirical models and model calculations
Supramolecular interactions play in molecular materials an important and non-trivial role. The optimization of the desired properties at the molecular level is per se a complex task that requires a synergic interplay between interpretative and synthetic abilities. But the optimization of material properties unavoidably relies on a thorough comprehension of supramolecular structure–properties relationships. Supramolecular interactions originate in molecular materials collective behavior: the material properties deviate quantitatively, and often also qualitatively from the additive (i.e. linear) response of isolated molecules. At the same time, cooperative behavior can appear and phenomena can show up with no counterpart at the molecular level. In push–pull chromophores, an electron donor (D) and an acceptor (A) group are joined by a -conjugated bridge. These molecules are interesting for second-order NLO applications [1], are good two-photon absorbers [2], and are widely studied in solution for their impressive solvatochromism [3]. They are also good emitters, with use in dye-lasers and in organic-LEDs [4], and a few of these structures behave as molecular rectifiers [5]. Push–pull chromophores are polar molecules, but, in view of their largely conjugated skeleton are also largely (hyper-)polarizable. Electrostatic interactions are therefore particularly important in samples with a medium-large concentration of push–pull chromophores: each molecule feels in fact the electric field generated by the surrounding (polar) molecules and, in view of its large polarizability, readjusts accordingly ∗
Corresponding author. Tel.: +39-0521-905461; fax: +39-0521-905556. E-mail address: [email protected] (A. Painelli).
its polarity, affecting in turn the local field. Collective and cooperative behavior then results as a consequence of the self-consistent interaction of each (polar and polarizable) chromophore with the surrounding chromophores. Push–pull chromophores resonate between a neutral (N) and a zwitterionic (I) structure, and their low-energy properties are well described in terms of a model that only accounts for two electronic basis states, |DA and |D+ A− , corresponding to the two limiting structures [6]. These two states, separated √ by an energy gap 2z0 , are mixed by a matrix element − 2t, yielding a ground state (GS) and an excited state. Both states are defined by a single parameter, ρ, intermediate between 0 and 1, measuring the weight of |D+ A− in the GS, and hence proportional to the GS polarity. 1 − ρ measures the polarity of the excited state [7]. The two-state model for the electronic structure is very simple, but it captures the relevant physics of push–pull chromophores. When extended to account for the coupling of electronic, vibrational and solvation degrees of freedom [8], it nicely describes steady-state and time-resolved electronic and vibrational spectra of push–pull chromophores in solution [9–12]. To investigate supramolecular interactions, we consider a cluster of (non-overlapping) push–pull chromophores only interacting via electrostatic forces as follows: H= (2z0 ρˆ i − σˆ xi ) + (1) Vij ρˆ i ρˆ j i
i>j
where the first term describes the molecular problem (we fix √ 2t as the energy unit) and the second term accounts for electrostatic interactions. The ρˆ i operator is the ionicity operator measuring the polarity of the ith site, ρˆ i = (1− σˆ zi )/2,
0379-6779/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0379-6779(03)00262-5
780
A. Painelli, F. Terenziani / Synthetic Metals 139 (2003) 779–781
Fig. 1. The two supramolecular arrangements discussed in this paper. The arrows schematically represent polar molecules.
with σˆ xi , σˆ zi representing the Pauli matrices. Vij measures the electrostatic interaction between two zwitterionic chromophores on sites i, j. Whereas the above Hamiltonian is fairly general, here we only discuss the two one-dimensional clusters sketched in Fig. 1. We impose periodic boundary conditions and assume unscreened interactions. Since dipolar interactions are unrealistic for push–pull chromophores, we assign each chromophore a fixed effective length l. The Vij in Eq. (1) are then fixed by v = e2 / l and w = l/r, with r measuring the interchromophore distance. For clusters with N sites, the Hamiltonian matrix is easily written and diagonalized on the 2N basis obtained from the direct product of the two basis states on each site. By exploiting translational symmetry we calculate the lowest eigenstates for clusters with up to 16 sites. In A-type clusters electrostatic intermolecular interactions disfavor charge separation: panel A in Fig. 2 shows that with decreasing interchromophore distance (increasing w), the average polarity of molecular sites (ρ = ρˆ i ) decreases. In particular, molecules with a large zwitterionic (I, ρ > 0.5) character in the non-interacting limit (w = 0), can be tuned towards a largely neutral (N, ρ < 0.5) GS for large enough interactions. Just the opposite occurs for lattice B, where supramolecular interactions favor on-site charge separation and ρ can be tuned from N to I with increasing w. Results in Fig. 2 are √ obtained by imposing v = 1 that corresponds, for typical 2t ∼ 1 eV relevant to push–pull chromophores [11], to a dipole length l ∼ 15 Å.
Fig. 2. Ground state polarity, ρ vs. the inverse interchromophore distance, w, for linear clusters of 16 sites with v = 1 and the two molecular arrangements sketched in the figure. (A) z0 = −1; (B) z0 = 1. Continuous and dashed lines show exact and mean-field results, respectively.
Data in Fig. 2 then show that the I to N crossover for A-lattices and the N to I crossover for B-lattices occur at intermolecular distances of ∼10 and 20 Å, respectively: the inversion of polarity from the isolated molecule to the clusters does not require extreme conditions, but is expected in samples with a medium-large concentration of chromophores. As a matter of fact, the variation of the molecular polarity with the environment is a fairly obvious phenomenon, and, as far as the effect of the solvent is concerned, it was discussed since date [3]. However, common approaches to the properties of materials with a large concentration of push–pull chromophores usually disregard this phenomenon [13]. Recently, Ashwell and Gandolfo demonstrated that the polarity of a cationic push–pull chromophore in a Langmuir–Blodgett film can be tuned through the N–I interface by the displacement of the counter-ion along the molecular axis [14]. We suggest that the molecular polarity can be more generally tuned by supramolecular interactions also in sample containing globally neutral, but polar and polarizable molecular units. The diagonalization of the Hamiltonian in Eq. (1) is difficult for large clusters. As far as GS properties are concerned, approximate treatments, based on a mean-field (MF) approximation, can be considered. In MF, the supramolecular problem reduces to the problem of a single molecule experiencing the electric field generated by the surrounding molecules [15,16]. Each molecule is then described by the same two-state model as the isolated molecule, but with z0 renormalized to z0 + Mρ, where M = i =j Vij /2. The resulting self-consistent two-state problem is easily solved, and yields ρ values in reasonable agreement with exact results (cf. Fig. 2). In attractive (B-type) lattices, for large z0 (and hence large |M|) values, the MF treatment predicts the appearance of large regions of bistability at the N–I interface, that correspond to the appearance of a discontinuous phase transition [15]. In this region, the material properties are unusual and deserve a detailed discussion which is beyond the scope of the present paper. At the molecular level, it is well known that the static NLO responses of push–pull chromophores strongly depend on the molecular polarity, ρ ([17] and references therein). Data in Fig. 2 then immediately suggest large supramolecular effects on static NLO responses. The simplest approach to static susceptibilities relies on an oriented gas picture: the responses of the cluster are calculated in this approach as the sum of contributions from a collection of molecules in the same geometrical arrangement and with the same mean-field polarity ρ they have in the material. The resulting estimates of the linear polarizability (α), first and second hyperpolarizabilities (β and γ, respectively) are reported in Fig. 3 (dotted lines). Exact susceptibilities (continuous lines in Fig. 3) are obtained from the successive derivatives of the dipole moment of the cluster (as obtained from the exact eigenstates) on the applied field. The deviation of the MF-oriented gas results from the exact curves is very large, and somewhat surprising, if contrasted with the good agreement between
A. Painelli, F. Terenziani / Synthetic Metals 139 (2003) 779–781
781
well calculated within the MF approximation, provided the collective behavior of the material is properly accounted for. Then the large deviations between the oriented gas estimates of static susceptibilities and the corresponding values obtained from the derivatives of the MF dipole moment are a clear signature of collective behavior, and are related to the non-linear response of polarizable chromophores to the perturbation induced by the surrounding chromophores.
Acknowledgements We thank italian MIUR for support and Z.G. Soos for useful discussions.
References Fig. 3. Static linear and non-linear polarizabilities vs. the inverse intermolecular distance. Results in the left and right panels are obtained for the same parameters as for the corresponding panels in Fig. 2.
MF and exact estimates of the GS polarity (cf. Fig. 2). As a matter of fact the oriented gas approach to the susceptibility fails since the response of a molecule to the applied field is different for the isolated molecule and for the molecule within the cluster. And in fact the MF approximation can lead to fairly accurate estimates of the susceptibility provided the oriented gas approximation is relaxed. Dashed lines in Fig. 3 show the susceptibility calculated from the successive derivatives of the GS dipole moment of the cluster, calculated in the MF approximation. The dipole moment of the cluster is indeed the sum of the (oriented) dipole moments of each molecule, but its derivatives are very different from the sum of the corresponding derivatives. Apart from deviations that occur just at the N–I crossover, the MF estimates of the susceptibilities nicely approximate the exact results: Static susceptibilities are GS properties and are reasonably
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12]
[13] [14] [15] [16] [17]
S.R. Marder, et al., Nature 388 (1997) 845. M. Rumi, et al., J. Am. Chem. Soc. 122 (2000) 9500. C. Reichardt, Chem. Rev. 94 (1994) 2319. M.A. Baldo, M.E. Thompson, S.R. Forrest, Nature 403 (2000) 750. R.M. Metzger, et al., J. Am. Chem. Soc. 119 (1997) 10455. J.C. Oudar, D.S. Chemla, J. Chem. Phys. 66 (1977) 2664. R.S. Mulliken, J. Am. Chem. Soc. 74 (1952) 811. A. Painelli, Chem. Phys. 245 (1999) 183. A. Painelli, F. Terenziani, Chem. Phys. Lett. 312 (1999) 211. A. Painelli, F. Terenziani, J. Phys. Chem. A 104 (2000) 11041; F. Terenziani, A. Painelli, D. Comoretto, J. Phys. Chem. A 104 (2000) 11049. B. Boldrini, et al., J. Phys. Chem. A 106 (2002) 6286. F. Terenziani, A. Painelli, in: V.M. Agranovich, G. C. La Rocca (Eds.), Proceedings of the International School of Physics Enrico Fermi, Course CXLIX, Organic Nanostructures: Science and Applications, IOS Press, Amsterdam, 2002, p. 569. Y.V. Pereverzev, O.V. Prezhdo, L.R. Dalton, Chem. Phys. Lett. 340 (2001) 328. G.J. Ashwell, D. Gandolfo, J. Mater. Chem. 12 (2002) 411. Z.G. Soos, et al., Ann. N. Y. Acad. Sci. 313 (1978) 442. E.V. Tsiper, Z.G. Soos, Phys. Rev. B 64 (2001) 195124. A. Painelli, Chem. Phys. Lett. 285 (1998) 208.
Excitonic and ultraexcitonic effects in supramolecular architectures of polar and polarizable chromophores Anna Painelli∗ , Francesca Terenziani Dip. Chimica GIAF, Università di Parma, viale delle Scienze 17/a, 43100 Parma, Italy, and INSTM-UdR Parma
Abstract Supramolecular interactions in clusters of polar and polarizable molecules are discussed. With reference to two different one-dimensional arrays of molecules we investigate the dependence of the molecular polarity and of static non-linear optical responses on the intermolecular distance. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Organic semiconductors based on conjugated molecules; Semi-empirical models and model calculations
Supramolecular interactions play in molecular materials an important and non-trivial role. The optimization of the desired properties at the molecular level is per se a complex task that requires a synergic interplay between interpretative and synthetic abilities. But the optimization of material properties unavoidably relies on a thorough comprehension of supramolecular structure–properties relationships. Supramolecular interactions originate in molecular materials collective behavior: the material properties deviate quantitatively, and often also qualitatively from the additive (i.e. linear) response of isolated molecules. At the same time, cooperative behavior can appear and phenomena can show up with no counterpart at the molecular level. In push–pull chromophores, an electron donor (D) and an acceptor (A) group are joined by a -conjugated bridge. These molecules are interesting for second-order NLO applications [1], are good two-photon absorbers [2], and are widely studied in solution for their impressive solvatochromism [3]. They are also good emitters, with use in dye-lasers and in organic-LEDs [4], and a few of these structures behave as molecular rectifiers [5]. Push–pull chromophores are polar molecules, but, in view of their largely conjugated skeleton are also largely (hyper-)polarizable. Electrostatic interactions are therefore particularly important in samples with a medium-large concentration of push–pull chromophores: each molecule feels in fact the electric field generated by the surrounding (polar) molecules and, in view of its large polarizability, readjusts accordingly ∗
Corresponding author. Tel.: +39-0521-905461; fax: +39-0521-905556. E-mail address: [email protected] (A. Painelli).
its polarity, affecting in turn the local field. Collective and cooperative behavior then results as a consequence of the self-consistent interaction of each (polar and polarizable) chromophore with the surrounding chromophores. Push–pull chromophores resonate between a neutral (N) and a zwitterionic (I) structure, and their low-energy properties are well described in terms of a model that only accounts for two electronic basis states, |DA and |D+ A− , corresponding to the two limiting structures [6]. These two states, separated √ by an energy gap 2z0 , are mixed by a matrix element − 2t, yielding a ground state (GS) and an excited state. Both states are defined by a single parameter, ρ, intermediate between 0 and 1, measuring the weight of |D+ A− in the GS, and hence proportional to the GS polarity. 1 − ρ measures the polarity of the excited state [7]. The two-state model for the electronic structure is very simple, but it captures the relevant physics of push–pull chromophores. When extended to account for the coupling of electronic, vibrational and solvation degrees of freedom [8], it nicely describes steady-state and time-resolved electronic and vibrational spectra of push–pull chromophores in solution [9–12]. To investigate supramolecular interactions, we consider a cluster of (non-overlapping) push–pull chromophores only interacting via electrostatic forces as follows: H= (2z0 ρˆ i − σˆ xi ) + (1) Vij ρˆ i ρˆ j i
i>j
where the first term describes the molecular problem (we fix √ 2t as the energy unit) and the second term accounts for electrostatic interactions. The ρˆ i operator is the ionicity operator measuring the polarity of the ith site, ρˆ i = (1− σˆ zi )/2,
0379-6779/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0379-6779(03)00262-5
780
A. Painelli, F. Terenziani / Synthetic Metals 139 (2003) 779–781
Fig. 1. The two supramolecular arrangements discussed in this paper. The arrows schematically represent polar molecules.
with σˆ xi , σˆ zi representing the Pauli matrices. Vij measures the electrostatic interaction between two zwitterionic chromophores on sites i, j. Whereas the above Hamiltonian is fairly general, here we only discuss the two one-dimensional clusters sketched in Fig. 1. We impose periodic boundary conditions and assume unscreened interactions. Since dipolar interactions are unrealistic for push–pull chromophores, we assign each chromophore a fixed effective length l. The Vij in Eq. (1) are then fixed by v = e2 / l and w = l/r, with r measuring the interchromophore distance. For clusters with N sites, the Hamiltonian matrix is easily written and diagonalized on the 2N basis obtained from the direct product of the two basis states on each site. By exploiting translational symmetry we calculate the lowest eigenstates for clusters with up to 16 sites. In A-type clusters electrostatic intermolecular interactions disfavor charge separation: panel A in Fig. 2 shows that with decreasing interchromophore distance (increasing w), the average polarity of molecular sites (ρ = ρˆ i ) decreases. In particular, molecules with a large zwitterionic (I, ρ > 0.5) character in the non-interacting limit (w = 0), can be tuned towards a largely neutral (N, ρ < 0.5) GS for large enough interactions. Just the opposite occurs for lattice B, where supramolecular interactions favor on-site charge separation and ρ can be tuned from N to I with increasing w. Results in Fig. 2 are √ obtained by imposing v = 1 that corresponds, for typical 2t ∼ 1 eV relevant to push–pull chromophores [11], to a dipole length l ∼ 15 Å.
Fig. 2. Ground state polarity, ρ vs. the inverse interchromophore distance, w, for linear clusters of 16 sites with v = 1 and the two molecular arrangements sketched in the figure. (A) z0 = −1; (B) z0 = 1. Continuous and dashed lines show exact and mean-field results, respectively.
Data in Fig. 2 then show that the I to N crossover for A-lattices and the N to I crossover for B-lattices occur at intermolecular distances of ∼10 and 20 Å, respectively: the inversion of polarity from the isolated molecule to the clusters does not require extreme conditions, but is expected in samples with a medium-large concentration of chromophores. As a matter of fact, the variation of the molecular polarity with the environment is a fairly obvious phenomenon, and, as far as the effect of the solvent is concerned, it was discussed since date [3]. However, common approaches to the properties of materials with a large concentration of push–pull chromophores usually disregard this phenomenon [13]. Recently, Ashwell and Gandolfo demonstrated that the polarity of a cationic push–pull chromophore in a Langmuir–Blodgett film can be tuned through the N–I interface by the displacement of the counter-ion along the molecular axis [14]. We suggest that the molecular polarity can be more generally tuned by supramolecular interactions also in sample containing globally neutral, but polar and polarizable molecular units. The diagonalization of the Hamiltonian in Eq. (1) is difficult for large clusters. As far as GS properties are concerned, approximate treatments, based on a mean-field (MF) approximation, can be considered. In MF, the supramolecular problem reduces to the problem of a single molecule experiencing the electric field generated by the surrounding molecules [15,16]. Each molecule is then described by the same two-state model as the isolated molecule, but with z0 renormalized to z0 + Mρ, where M = i =j Vij /2. The resulting self-consistent two-state problem is easily solved, and yields ρ values in reasonable agreement with exact results (cf. Fig. 2). In attractive (B-type) lattices, for large z0 (and hence large |M|) values, the MF treatment predicts the appearance of large regions of bistability at the N–I interface, that correspond to the appearance of a discontinuous phase transition [15]. In this region, the material properties are unusual and deserve a detailed discussion which is beyond the scope of the present paper. At the molecular level, it is well known that the static NLO responses of push–pull chromophores strongly depend on the molecular polarity, ρ ([17] and references therein). Data in Fig. 2 then immediately suggest large supramolecular effects on static NLO responses. The simplest approach to static susceptibilities relies on an oriented gas picture: the responses of the cluster are calculated in this approach as the sum of contributions from a collection of molecules in the same geometrical arrangement and with the same mean-field polarity ρ they have in the material. The resulting estimates of the linear polarizability (α), first and second hyperpolarizabilities (β and γ, respectively) are reported in Fig. 3 (dotted lines). Exact susceptibilities (continuous lines in Fig. 3) are obtained from the successive derivatives of the dipole moment of the cluster (as obtained from the exact eigenstates) on the applied field. The deviation of the MF-oriented gas results from the exact curves is very large, and somewhat surprising, if contrasted with the good agreement between
A. Painelli, F. Terenziani / Synthetic Metals 139 (2003) 779–781
781
well calculated within the MF approximation, provided the collective behavior of the material is properly accounted for. Then the large deviations between the oriented gas estimates of static susceptibilities and the corresponding values obtained from the derivatives of the MF dipole moment are a clear signature of collective behavior, and are related to the non-linear response of polarizable chromophores to the perturbation induced by the surrounding chromophores.
Acknowledgements We thank italian MIUR for support and Z.G. Soos for useful discussions.
References Fig. 3. Static linear and non-linear polarizabilities vs. the inverse intermolecular distance. Results in the left and right panels are obtained for the same parameters as for the corresponding panels in Fig. 2.
MF and exact estimates of the GS polarity (cf. Fig. 2). As a matter of fact the oriented gas approach to the susceptibility fails since the response of a molecule to the applied field is different for the isolated molecule and for the molecule within the cluster. And in fact the MF approximation can lead to fairly accurate estimates of the susceptibility provided the oriented gas approximation is relaxed. Dashed lines in Fig. 3 show the susceptibility calculated from the successive derivatives of the GS dipole moment of the cluster, calculated in the MF approximation. The dipole moment of the cluster is indeed the sum of the (oriented) dipole moments of each molecule, but its derivatives are very different from the sum of the corresponding derivatives. Apart from deviations that occur just at the N–I crossover, the MF estimates of the susceptibilities nicely approximate the exact results: Static susceptibilities are GS properties and are reasonably
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12]
[13] [14] [15] [16] [17]
S.R. Marder, et al., Nature 388 (1997) 845. M. Rumi, et al., J. Am. Chem. Soc. 122 (2000) 9500. C. Reichardt, Chem. Rev. 94 (1994) 2319. M.A. Baldo, M.E. Thompson, S.R. Forrest, Nature 403 (2000) 750. R.M. Metzger, et al., J. Am. Chem. Soc. 119 (1997) 10455. J.C. Oudar, D.S. Chemla, J. Chem. Phys. 66 (1977) 2664. R.S. Mulliken, J. Am. Chem. Soc. 74 (1952) 811. A. Painelli, Chem. Phys. 245 (1999) 183. A. Painelli, F. Terenziani, Chem. Phys. Lett. 312 (1999) 211. A. Painelli, F. Terenziani, J. Phys. Chem. A 104 (2000) 11041; F. Terenziani, A. Painelli, D. Comoretto, J. Phys. Chem. A 104 (2000) 11049. B. Boldrini, et al., J. Phys. Chem. A 106 (2002) 6286. F. Terenziani, A. Painelli, in: V.M. Agranovich, G. C. La Rocca (Eds.), Proceedings of the International School of Physics Enrico Fermi, Course CXLIX, Organic Nanostructures: Science and Applications, IOS Press, Amsterdam, 2002, p. 569. Y.V. Pereverzev, O.V. Prezhdo, L.R. Dalton, Chem. Phys. Lett. 340 (2001) 328. G.J. Ashwell, D. Gandolfo, J. Mater. Chem. 12 (2002) 411. Z.G. Soos, et al., Ann. N. Y. Acad. Sci. 313 (1978) 442. E.V. Tsiper, Z.G. Soos, Phys. Rev. B 64 (2001) 195124. A. Painelli, Chem. Phys. Lett. 285 (1998) 208.